# Properties

 Label 20.5.d.a Level 20 Weight 5 Character orbit 20.d Self dual yes Analytic conductor 2.067 Analytic rank 0 Dimension 1 CM discriminant -20 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 20.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.06739926168$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} + 82q^{7} - 64q^{8} - 77q^{9} + O(q^{10})$$ $$q - 4q^{2} + 2q^{3} + 16q^{4} + 25q^{5} - 8q^{6} + 82q^{7} - 64q^{8} - 77q^{9} - 100q^{10} + 32q^{12} - 328q^{14} + 50q^{15} + 256q^{16} + 308q^{18} + 400q^{20} + 164q^{21} - 878q^{23} - 128q^{24} + 625q^{25} - 316q^{27} + 1312q^{28} - 1198q^{29} - 200q^{30} - 1024q^{32} + 2050q^{35} - 1232q^{36} - 1600q^{40} + 482q^{41} - 656q^{42} - 2078q^{43} - 1925q^{45} + 3512q^{46} + 4402q^{47} + 512q^{48} + 4323q^{49} - 2500q^{50} + 1264q^{54} - 5248q^{56} + 4792q^{58} + 800q^{60} - 4078q^{61} - 6314q^{63} + 4096q^{64} - 4478q^{67} - 1756q^{69} - 8200q^{70} + 4928q^{72} + 1250q^{75} + 6400q^{80} + 5605q^{81} - 1928q^{82} + 8002q^{83} + 2624q^{84} + 8312q^{86} - 2396q^{87} + 4322q^{89} + 7700q^{90} - 14048q^{92} - 17608q^{94} - 2048q^{96} - 17292q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0
−4.00000 2.00000 16.0000 25.0000 −8.00000 82.0000 −64.0000 −77.0000 −100.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.5.d.a 1
3.b odd 2 1 180.5.f.b 1
4.b odd 2 1 20.5.d.b yes 1
5.b even 2 1 20.5.d.b yes 1
5.c odd 4 2 100.5.b.b 2
8.b even 2 1 320.5.h.a 1
8.d odd 2 1 320.5.h.b 1
12.b even 2 1 180.5.f.a 1
15.d odd 2 1 180.5.f.a 1
20.d odd 2 1 CM 20.5.d.a 1
20.e even 4 2 100.5.b.b 2
40.e odd 2 1 320.5.h.a 1
40.f even 2 1 320.5.h.b 1
60.h even 2 1 180.5.f.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.d.a 1 1.a even 1 1 trivial
20.5.d.a 1 20.d odd 2 1 CM
20.5.d.b yes 1 4.b odd 2 1
20.5.d.b yes 1 5.b even 2 1
100.5.b.b 2 5.c odd 4 2
100.5.b.b 2 20.e even 4 2
180.5.f.a 1 12.b even 2 1
180.5.f.a 1 15.d odd 2 1
180.5.f.b 1 3.b odd 2 1
180.5.f.b 1 60.h even 2 1
320.5.h.a 1 8.b even 2 1
320.5.h.a 1 40.e odd 2 1
320.5.h.b 1 8.d odd 2 1
320.5.h.b 1 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{5}^{\mathrm{new}}(20, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T$$
$3$ $$1 - 2 T + 81 T^{2}$$
$5$ $$1 - 25 T$$
$7$ $$1 - 82 T + 2401 T^{2}$$
$11$ $$( 1 - 121 T )( 1 + 121 T )$$
$13$ $$( 1 - 169 T )( 1 + 169 T )$$
$17$ $$( 1 - 289 T )( 1 + 289 T )$$
$19$ $$( 1 - 361 T )( 1 + 361 T )$$
$23$ $$1 + 878 T + 279841 T^{2}$$
$29$ $$1 + 1198 T + 707281 T^{2}$$
$31$ $$( 1 - 961 T )( 1 + 961 T )$$
$37$ $$( 1 - 1369 T )( 1 + 1369 T )$$
$41$ $$1 - 482 T + 2825761 T^{2}$$
$43$ $$1 + 2078 T + 3418801 T^{2}$$
$47$ $$1 - 4402 T + 4879681 T^{2}$$
$53$ $$( 1 - 2809 T )( 1 + 2809 T )$$
$59$ $$( 1 - 3481 T )( 1 + 3481 T )$$
$61$ $$1 + 4078 T + 13845841 T^{2}$$
$67$ $$1 + 4478 T + 20151121 T^{2}$$
$71$ $$( 1 - 5041 T )( 1 + 5041 T )$$
$73$ $$( 1 - 5329 T )( 1 + 5329 T )$$
$79$ $$( 1 - 6241 T )( 1 + 6241 T )$$
$83$ $$1 - 8002 T + 47458321 T^{2}$$
$89$ $$1 - 4322 T + 62742241 T^{2}$$
$97$ $$( 1 - 9409 T )( 1 + 9409 T )$$